Theorem opelvvg 4768

Description: Ordered pair membership in the universal class of ordered pairs. (Contributed by Mario Carneiro, 3-May-2015.)

Assertion

Ref	Expression
opelvvg	|- ( ( A e. V /\ B e. W ) -> <. A , B >. e. ( _V X. _V ) )

Proof of Theorem opelvvg

Step	Hyp	Ref	Expression
1		elex 2811	. 2 |- ( A e. V -> A e. _V )
2		elex 2811	. 2 |- ( B e. W -> B e. _V )
3		opelxpi 4751	. 2 |- ( ( A e. _V /\ B e. _V ) -> <. A , B >. e. ( _V X. _V ) )
4	1, 2, 3	syl2an 289	1 |- ( ( A e. V /\ B e. W ) -> <. A , B >. e. ( _V X. _V ) )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2200   _Vcvv 2799   <.cop 3669   X. cxp 4717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-opab 4146  df-xp 4725

This theorem is referenced by:  relsnopg  4823  opvtxfv  15823  opiedgfv  15826